0
votes
0answers
14 views

$\ K = \{ (1,1) , (2,1) , (3,1) \} f(R) = RK$

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ is f injective? ...
1
vote
1answer
16 views

Solving a poset for less than equal?

I don't completely understand posets yet, so I'm confused on how to do this particular problem. Here is the question: Let S be the set of all real numbers. Prove that the less than or equal to ...
1
vote
3answers
53 views

Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
1
vote
1answer
28 views

Proving a Bound for Oddtown-Eventown or Clubtown

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of ...
0
votes
2answers
30 views

Writing in set builder notation

I need to write following set in to set builder notation. $\{ 0,3,6,9,12 \}$ Solution which I found on internet is: $\{3x\;|\;\text{where }x\text{ is an integer and }0\leq x \leq 4\}$ What I ...
1
vote
3answers
46 views

Understanding Set Theory and Proving $A \cap(B\cup A) = A$

I am trying to wrap my head around discrete mathematics in order to help my understanding of self taught programming. I am now trying to understand Set Theory, more specifically proving certain ...
0
votes
0answers
28 views

Proving a set is equal to another set

For all sets $A$ and $B,(B-A)=B\cap A^C$. I would like to know if this proof is correct or if I am on the right track. Here it is: Let $b \in B$ such that $b \notin A$ than $b \in B$ and $b \in ...
1
vote
2answers
32 views

How to Prove this Set Question? [closed]

For both sets C and D, provide a proof that C ∪ (D − /C) = C "/C" is a set's complement of C
-1
votes
2answers
103 views

Give a Bijection that is in-between 2 intervals and use a formal proof to show that it is a bijection. [duplicate]

∀w,x,y,z ∈ R, w < x and y < z. Given that information, supply a bijection between the two intervals. (w,x) and (y,z) Then after you find the bijection, provide a formal proof that what you found ...
0
votes
1answer
52 views

Can you conclude that A = B if A, B, and C are sets such that…

a. A ∪ C = B ∪ C b. A ∩ C = B ∩ C c. A ∩ C = B ∩ C and A ∪ C = B ∪ C My method of solving this was to convert everything to propositional logic, then to solve it to show that none of the above are ...
1
vote
1answer
41 views

$M_{R^n}$; how to derive $n$ for transitive closure?

When finding the transitive closure of a relation $R$, I convert $R$ into a boolean matrix $M_R$, and find the union between $M_R$ and its powers up to $n$. $$M_{R^*} = M_{R^1} \lor M_{R^2} \lor ...
0
votes
1answer
39 views

set theory, infinite set proof, is it alright?

$\Bbb{N}$ is the natural numbers set (included $0$). let be $n\in\Bbb{N}$, $A_n = \{x\in \Bbb{N}|0\leq x \leq n\}$ prove of disprove: $$\forall n,k \in \Bbb{N},\exists m \in\Bbb{N}(|A_m - A_n|=k)$$ ...
0
votes
1answer
35 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
0
votes
1answer
16 views

Find two elements x,y in P(A) such that $xy=0, x\ne0$ and $ y\ne0$.

The question is as follows: Let $A = \{a, b\}$ and list the four elements of the power set $P (A)$. We consider the operations $+$ to be $\cap$, · to be $\cup$, and complement to be set complement. ...
1
vote
0answers
17 views

Explain why the description defines a Boolean Algebra

This is the exercise: Let A = {a,b} and list the four elements of the power set P(A). We consider the operations + to be ∪, . to be ∩, and complement to be set complement. Consider 1 to be A and 0 to ...
0
votes
1answer
30 views

Can an element hood test be converted into an existential statement?

I'm just curious whether it makes sense to convert a statement of the form: $$ y\in \{x\in A : \phi(x) \} \;\; \text{into the form} \;\; \exists x(\,...) $$ It's just that in the book I'm reading the ...
0
votes
0answers
66 views

for n∈N A(n)={x∈N | 0<= x <= n}, prove the following statements

please help me improve this proofes, or find a more formal mathematical version of them. N is the set of natural numbers N = {0,1,2,...,} for all $n∈N$, there is $A(n) = \{x∈N | 0\le x \le n\}$ ...
0
votes
2answers
50 views

understanding reflexive transitive closure

Suppose I have the following relation $$R = \{(1,1), (2,3), (3,1)\}$$ To make it reflexive we add the following missing pairs: $$ \{(2,2), (3,3)\}$$ Now I wonder how to find the reflexive transitive ...
1
vote
1answer
43 views

did I solves this sets correctly?

need to solve this only with what we know about 'algebra of sets', is everythink I did is legal and it's correct? i. (A1$\bigcup$A2) $\setminus$ (B1$\bigcap$B2) = (A1 $\setminus$ B1)$\bigcup$(A1 ...
1
vote
0answers
36 views

Prove the following sets equalities

I'm really struggling with proofes, please tell me if I'm correct and if there is a better way to prove (or disprove) the following: i) $(A \setminus B) \setminus B = A \setminus B$ My answer: ...
0
votes
0answers
30 views

are those sets regarding the empty set are true or false?

for X = $\emptyset$ Y = {$\emptyset$ , foo} (foo is an element which is not a set) Z = { $\emptyset$ , {$\emptyset$} } are those sentences true or false? I wrote my answer next to them, please ...
1
vote
1answer
31 views

solve a set in 'algerba of sets' way.

two question, need to solve each in 'algebra of sets' way $$(A \cup B) - (C \cap D) = (A - C) \cup (A - D) \cup (B - C) \cup (B - D)$$ $$(A \cup B) \cap (C \cap D)^\complement = (A \cap ...
0
votes
2answers
55 views

T and F on some discrete math concepts

I was studying and these questions came up on a review guide on the inter webs, but could was wondering if I was correct on them. 1.Let $B$ $\subset$ $A$ and $f$ : $B$ $\subset$ $A$ be a 1-1 and ...
0
votes
2answers
65 views

How can you infer that $A\cap B = \emptyset$?

Given: $$H:((A \cup B) \to \{ 0,1\} ) \to ((A \to \{ 0,1\} ) \times (B \to \{ 0,1\} ))$$ $$H = \lambda f \in (A \cup B) \to \{ 0,1\} .\left\langle {\lambda a \in A.f(a),\lambda b \in B.f(b)} ...
0
votes
2answers
43 views

Cartesian Product for not a finite number of elements has how many elements

Suppose P is a set that has m elements and Q is a set with n elements. How many elements will their Cartesian product, PxQ have?
0
votes
1answer
46 views

Proving properties of the images and inverse images of functions

Let $f : X\to Y; \;\;g : Y\to Z;\,$ and $\;g \circ f : X\to Z.$ Prove or disprove a) For all subsets $\,A \subseteq X,\;\; f^{-1}(f(A)) = A$. b) For all subsets $\,B \subseteq Y,\;\; f(f^{-1}(B)) ...
0
votes
1answer
33 views

Converting English events to Set Theory Notation using Operators

Just having a little bit of a hard time converting these 3 English statements into proper set-theory notation. They are just very ambigious. A, B, C, D are events in sample space S. Express the ...
0
votes
1answer
16 views

Are all relations that are symmetric and anti-symmetric a subset of the reflexive relation?

Simple question, but I can't seem to find a guaranteed answer. A symmetric set contains (a, b) if it contains (b, a), but an ...
0
votes
1answer
46 views

Help understanding a theorem on transitivity of a relation

The theorem states this: The relation R on a set A is transitive if and only if $R^n \subseteq R$ for n = 1, 2, 3,... What I'm reading is that the nth power of that set is transitive if the set ...
0
votes
1answer
60 views

More transcendental numbers than natural numbers [duplicate]

Are there any simple proofs that obtain this result? I haven't been able to find one online.
2
votes
3answers
70 views

Show that the set of all infinite subsets of $\mathbb{N}$ has cardinality greater than $\mathbb{N}$

Is there any way to solve this problem using the diagonalization method? I know there's the proof that uses $\mathcal{P}(\mathbb{N})$ = set of all finite subsets $\cup$ set of all infinite subsets, ...
0
votes
2answers
65 views

Set Theory question show that

Would this be correct Let $A$ and $X$ be subsets of a universe $U$. Show that if $A \cup X =$ Universe $A \cap X = \emptyset$ then $X=A^c$ $A \cup X =$ Universe (given) $A \cup A^c =$ Universe ...
0
votes
1answer
40 views

Number of ways to choose two disjoint subsets of a set

Let $A$ be a set of $n$ elements. Then, in how many ways, can we choose an ordered pair $(B,C)$, where $B,C$ are disjoint subsets of $A$? Let A={1}, then B can be {phi} and C can be {1}. So, one ...
2
votes
5answers
57 views

Prove that $A\setminus (B\setminus C) = (A\setminus B) \cup (A\setminus C^c)$ for sets $A,\ B,\ C$ in some Universal Set $U$.

I'm working on this proof for some students I am tutoring and I've gotten a little stuck. I want to show them how to do a proof in complete, extravagant detail and get them familiar with ''element ...
3
votes
6answers
99 views

Prove that $(B - A) \cup (C - A) = (B \cup C) - A$ by showing that each side is a subset of the opposite side.

A big problem is that I never even know where to start with proofs. Then I panic and get absolutely nowhere. To reiterate: Prove that $(B - A) \cup (C - A) = (B \cup C) - A$ by showing that each side ...
1
vote
2answers
32 views

Sets, subset relations, and elements in a set

I can't quite get a handle on subsets. Are the following true or false? $ \{\{\emptyset\}\}\subset\{\emptyset,\{\emptyset\}\} $ $\{\{\emptyset\}\}\subset\{\{\emptyset\},\{\emptyset\}\}$ $\{x\} ...
3
votes
2answers
102 views

I'm not understanding what a reflexive set is

I'm not quite getting the concept of a reflexive set in my discrete math class. I think I understand that a reflexive set is the product of set $\mathit{A} \times \mathit{A}$ the idea that we're ...
1
vote
2answers
31 views

Intersections/Reunions of power sets

Let $P_i$ be the power set of $A_i=\{1,2,3,\cdots ,i\}$. What is: $$\bigcap_{i=1}^{n}(P_{i+1} - P_{i})$$ $$\bigcup_{i=1}^{n}(P_{i+1} - P_{i})$$ The problem asks you to find those two sets ...
0
votes
1answer
32 views

Constructing sets involving predicates. Let $P(x),Q(x)$ be predicates over a set $X$?

Let $X$ be a set and $P(x),Q(x)$ be predicates over $X$. Consider the sets $$Y = \{y\in X\mid P(y)\}$$ $$Z = \{z\in X\mid Q(z)\}$$ Complete the following sentences with quantified propositional ...
0
votes
2answers
36 views

How do you solve for the cardinality of a power set of some complex set? (i.e. $|\mathcal P(A^n)|$ , $|\mathcal P(A\cup B)|$ )

Suppose $A$ is some set such that $A = \{a_1,a_2,\dotsb,a_n\}$. We know that $|A|=n$. We know that $\mathcal P(A)= 2^n$. Now let $A^n$ denote the cartesian product of a set A with itself n times. ...
0
votes
1answer
39 views

$if An \subseteq A $ for all $n \in \mathbb{N}, $ then $ \bigcup_{n=1}^\infty An \subseteq A $

I was given this as an exercise in my discrete math class and I have been having a lot of trouble, I am not really sure how to approach a problem like this. Any help is appreciated!Thank you! (this is ...
1
vote
1answer
39 views

Prove $A $ \ $B $ = $A \cap B^c $

I see the use of $A $ \ $B $ = $A \cap B^c $ being used in bigger problems but how do you prove this? Is the proof as simple as: $A $ \ $B $ $\iff$ $ x \in (A \setminus B) \iff x\in A \cap ...
2
votes
2answers
54 views

Prove $(A^c)^c = A$

Hey guys I know this is a super easy example but, this is my first day doing this stuff and i really need to get the basics down. Is this how to go about proving $(A^c)^c = A$ $$ \begin{align} ...
0
votes
1answer
41 views

Reflexive relation on set of $n$ elements

How many reflexive relations are there on a set of $n$ elements? I did the problem and I got the answer $2 ^ {n ^ 2}$. Is it correct? Thanks for the help..!!
1
vote
1answer
123 views

Functions that are sets of all function - proofs

I'm going through the book Proofs and fundamentals, by Bloch, and it doesn't include a solution manual for it's examples. It doesn't have many examples on notation and proof strategy for certain ...
0
votes
2answers
19 views

Which $Y \subseteq \{a,b,c,d\}$ satisfy $Y - B = X - B$, where $B = \{b,d\}$ and $X = \{a,b\}$?

I am having a problem with my math assignment, and i'm honestly not even sure of where to begin with this one. The question is: Let $A=\{a,b,c,d\}$, $B = \{b,d\}$ and $X = \{a,b\}$. Determine all ...
2
votes
1answer
58 views

Bijective function proof in $R\times R$ and $Z\times N$

How can I verify if these functions are bijective? $ f_4:\Bbb{R^2} \rightarrow \Bbb{R^2}, \ (x,\ y)\mapsto (x+y,\ x-y)$ $ f_5:\Bbb{Z} \times \Bbb{N^*} \rightarrow \Bbb Q, \ (p,\ q)\mapsto p + ...
2
votes
2answers
45 views

find $P\{P\{0\}\}$. $P$ represents the power set.

I'm assuming that I'm trying to find the power set of a power set? I start from the inner power set, $P\{0\}$. $P\{0\}= \{ 0, \{0\} \}$. Now I do $P\{ 0, \{0\} \}$ which is $\{ 0, \{0\}, \{\{0\}\} ...
0
votes
2answers
106 views

Can someone please help with my inverse function and sets discrete math problem?

To save me some time writing everything out in latex, I'm adding a picture of the question and Ill try to explain what I understand for the problem. Just a heads up, I'm really not sure how to do this ...
0
votes
2answers
46 views

Boolean Logic - Why doesn't the Associativity Law work in this scenario?

By the associativity law, shouldn't the statement below be true? I understand that the truth tables are different but where exactly does the associativity law apply then if this is False? ...